The Regularity And Saturation Class For Linear Combinations Of Szász-Mirakjan Operators

The operator approximation is an important part of approximation theory. With the rapid development of technology, it is widely used in wavelet analysis, neural net-work, statistics and so on. This thesis mainly concerns the regularity of Szdsz-Mirakjan operators and the saturation class for linear combinations of Szasz-Mirakjan operators. The contents are as follows:In Chapter 1, we introduce the relationship between the Szasz-Mirakjan operators and the Bernstein operators, the related conclusions of Bernstein operators approximat-ing functions f(x) and coming the similar conclusions when Szdsz-Mirakjan operators approximates functions f(x). Moreover, the chapter also introduces some conclusions of the deformated Szdsz-Mirakjan operators approximating functions f(x).In Chapter 2, we study the regularity of Szdsz-Mirakjan operators. First, we con-struct a class of functions so as to satisfy the differential operators bounded. Then, we prove bounded of the highest order derivative item of differential operators. It shows the regularity of Szasz-Mirakjan operators.In Chapter 3, we study the saturation class for linear combinations of Szdsz-Mirakjan operators. The order of approximation can only reach the 1/n when Szdsz-Mirakjan operators approximates functions f(x). In order to improve the order of approximation, we consider the linear combinations of Szdsz-Mirakjan operators. It shows the saturation class for linear combinations of Szdsz-Mirakjan operators.